Not at all. Limbo was the place where people who couldn't go to heaven but shouldn't go to Hell go. (a baby who died before it was baptized, good people who lived before Jesus - if you can't go to heaven unless you believe in Jesus, how could you have believed in him BEFORE he was born? - those were the reasons for Limbo)
Purgatory was a place for those who were going to heaven - but had committed sins went to.
Say you have an old nun who did nothing but serve others for her entire life. She spends her life in prayer and praising Jesus. She dies and goes straight to heaven.
Now compare her to Adolph Hitler (or any other mass murderer or uber-evil person). He spent his life causing incredible suffering and misery for others. Then 60 seconds before Hitler dies, he sees the light and says, "I accept Jesus in my heart as my personal savior." Not just "says" it - but really and truly means it. He dies and also goes straight to heaven (because now he is saved).
That didn't seem fair. Why should an evil person who finds religion in the last few seconds of their life after a lifetime of ultimate evil get the same reward as the good nun?
That was the reason for Purgatory - yes, Hitler would go to heaven because he was saved before he died - BUT he still has to atone for the evil he did and he did that atonement in Purgatory.
I don't know what the Church's teaching is about purgatory anymore - but that was the original reason for it.
That answer is unbiblical. Sorry, but it just is. You can no more "atone for your sin" than you can sprout wings and fly to Mars for the weekend. Salvation is from God alone, and he doesn't need you or me screwing it all up with our "atoning."
Your point about it not being "fair" is very true. However, when you and I get to the judgment seat of God, I guarantee you we don't want "FAIR" we want mercy. "FAIR" means we all go straight to hell, with not a word to say in our defense (unlike here on earth, where men argue from moral ignorance of their own condition). We leave "fair" at the cross of Christ. It is mercy, grace and unearned favor. Nothing else.